Isn't "believing that God doesn't exist" the same thing as "not believing in God"?

Obviously, it is illogical. One can't dismiss this as "purely semantics". It is actually indicating a subtle, yet critical issue.

The difference is the level of assertion.

Not having a belief is not accepting a claim as true. If one actively believed there was no god, that would be accepting a claim as true. One who lacks a belief is waiting to find out what's actually true.

It is the difference between Innocent, Not Guilty and Guilty.
It is the difference between Positive, Neutral and Negative.
It is the difference between Confirmed, Pending and Falsified.

Consider the question of whether there's an even number or odd number of stars in the universe. If a person claims that there's an odd number of stars, we wouldn't believe that person's claim, due to lack of evidence. That, however, is not an assertion that there's an even number of stars in the universe. We can reject both claims as unsupported, and the status of the question is "unknown".

To say that not believing in a god is the same as believe there are no gods, is like saying that not believing there's an odd number of stars in the universe is the same as believing there's an even number.

Many atheists are "neutral", in that they are not necessarily accepting that a god absolutelydoes not exist, but rather are waiting for theists to demonstrate their claims, using the scientific method, preferably.

Colloquially, most people think of this position as "agnostic", but in fact, it is both. In short, agnosticism deals with what is knowable, and atheism deals with what's believed. Most self-described atheists describe themselves as "agnostic atheist".